a 2m long string of mass 10g is clamped at boths ends.the tension in the string is 150N.the string is plucked so that it oscillate.what is the wavelength and frequency of the resulting wave it is produce a standing wave with two antinodes?
To find the wavelength and frequency of the resulting wave that produces a standing wave with two antinodes, we need to use the equation that relates wave speed (v), wavelength (λ), and frequency (f):
v = λ * f
First, we need to calculate the wave speed. The wave speed is determined by the tension in the string and the linear mass density (μ). The linear mass density is the mass per unit length of the string.
μ = m / L
where,
μ = linear mass density of the string
m = mass of the string = 10g = 0.01kg
L = length of the string = 2m
μ = 0.01kg / 2m = 0.005 kg/m
The wave speed can be calculated using the formula:
v = √(T / μ)
where,
v = wave speed
T = tension in the string = 150N
μ = linear mass density of the string
v = √(150N / 0.005 kg/m) = √(30,000 m^2/s^2) = 173.21 m/s (rounded to two decimal places)
Since the standing wave has two antinodes, the distance between the two antinodes is equal to half a wavelength (λ/2).
Given the length of the string (L) is 2m, and the distance between the antinodes is λ/2, we have:
L = λ/2
Rearranging the equation, we can solve for the wavelength (λ):
λ = 2 * L
λ = 2 * 2m = 4m
So, the wavelength of the resulting wave is 4 meters.
Now, we can use the wave speed equation to find the frequency (f):
v = λ * f
Rearranging the equation, we can solve for frequency:
f = v / λ
f = 173.21 m/s / 4m = 43.30 Hz (rounded to two decimal places)
Therefore, the frequency of the resulting wave is 43.30 Hz, and the wavelength is 4 meters.